4. A surface is defined by \( z^{2}=4\left(x^{2}+y^{2}\right) \) where \( 0 \leq z \leq 6 \). If a vector field \( \mathbf{F}=z \mathbf{i}+x y^{2} \mathbf{j}+x^{2} z \mathbf{k} \) exists throughout and on the boundary circle \( C \), show that
\[
\oint_{C} \mathbf{F} \bullet \mathrm{~d} \mathbf{r}=\int_{S} \operatorname{curl}(\mathbf{F}) \bullet \mathrm{d} \mathbf{S} .
\]
[ This is a verification of Green's theorem.]
5. Verify the Gauss divergence theorem for \( \mathbf{F}=4 x \mathbf{i}-2 y^{2} \mathbf{j}+z^{2} \mathbf{k} \) over the region bounded by \( x^{2}+y^{2}=4, z=0 \) and \( z=3 \).